Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4449 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,18)(11,13,17,15)(12,20,21,16)(19,24,22,23)(25,26)(27,35)(28,32)(29,38)(30,34)(31,37)(33,40)(36,39);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,24)(20,23)(21,22)(25,26,30,34)(27,31,37,35)(28,36,39,32)(29,33,40,38);(1,3)(2,4)(5,6)(7,8)(12,21)(16,20)(19,22)(23,24)(25,27,30,37)(26,31,34,35)(28,33,39,38)(29,36,40,32);(3,6)(4,8)(9,12,14,21)(10,16,18,20)(11,19,17,22)(13,23,15,24)(25,27)(26,31)(28,38)(29,36)(30,37)(32,40)(33,39)(34,35);(1,4,5,8)(2,6,7,3)(9,11)(10,13)(12,22)(14,17)(15,18)(16,24)(19,21)(20,23)(25,28)(26,32)(27,33)(29,31)(30,39)(34,36)(35,40)(37,38);(1,2)(3,4)(5,7)(6,8)(9,13)(10,17)(11,18)(12,23)(14,15)(16,22)(19,20)(21,24)(25,29)(26,33)(27,36)(28,35)(30,40)(31,39)(32,37)(34,38)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,d^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9068257);